K-connectedness and the homotopy-lifting property

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In summary, K-connectedness is a concept in topology that describes the connectivity of a topological space. It relates to the homotopy-lifting property, which characterizes a space's ability to be continuously deformed into another space. While K-connectedness and the homotopy-lifting property have significance in mathematics, they also have practical applications in fields such as data analysis and computer science.
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Bacle
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Hi:
Is there a relation between k-connectedness ( meaning 1st, 2nd,..,k-th fundamental
groups are trivial.) and having the homotopy-lifting property.? . This may be
vaguely-related to being able to extend global sections from the j-th skeleton, to
the (j+1)-st skeleton (j<=k, obviously), but I am not sure.

How about k-connectedness for a pair (A,X) ( A a subspace of X).

Thanks.
 
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Could you be more specific? Are we fixing a triple, or are we asking about whether any surjective map is a fibration?
 

FAQ: K-connectedness and the homotopy-lifting property

1. What is the concept of K-connectedness in topology?

K-connectedness is a concept in topology that describes the connectivity of a topological space. A space is said to be K-connected if it cannot be separated into two non-empty subsets by removing K-1 points. In other words, there is no set of K-1 points whose removal disconnects the space.

2. How does K-connectedness relate to the homotopy-lifting property?

The homotopy-lifting property is a key property of topological spaces that characterizes their ability to be continuously deformed into one another. K-connectedness is a sufficient condition for a space to have the homotopy-lifting property. This means that if a space is K-connected, then it also has the homotopy-lifting property.

3. Can you give an example of a space that is K-connected but does not have the homotopy-lifting property?

Yes, a classic example is the Hawaiian earring space. This space is K-connected, as removing any finite number of points does not disconnect it. However, it does not have the homotopy-lifting property, as there exists a continuous map from a circle to the space that cannot be lifted to a continuous map from the disk to the space.

4. What is the significance of K-connectedness and the homotopy-lifting property in mathematics?

K-connectedness and the homotopy-lifting property play a fundamental role in algebraic topology, which is the study of topological spaces using algebraic tools. These concepts are used to classify and distinguish between different topological spaces, and they have applications in fields such as differential geometry and physics.

5. Are there any practical real-world applications of K-connectedness and the homotopy-lifting property?

While these concepts were originally developed in pure mathematics, they have also found applications in data analysis and computer science. For example, the homotopy-lifting property is used in topological data analysis to analyze and visualize high-dimensional data sets. Additionally, the concept of K-connectedness has been used in computer science to develop efficient algorithms for path planning and routing problems.

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